TEM images both of as-deposited and annealed thin films are shown in Figure 2. The annealed temperature was 873 K. In the pictures of the cross section of the multilayer, the dark parts reveal the Au layers. In the as-depo- sited thin film, the well-stacked Au and Si layers were observed (Figure 2(a)). By Fourier transform (FT) of atomic arrangement in Figure 2(a), the as-deposited thin film was found to be amorphous. On the other hand, the layered structure completely disappeared after annealing (Figure 2(b)). Instead, Au nanoparticles existed in the thin film, where the nanoparticles randomly locate inside Si. FT of the Au nanoparticles in Figure 2(b) suggests that the nanoparticles crystallized from amorphous by annealing. Thus, irreversible morphological transformation occurred accompanying by crystallization of the Au nanoparticles (c-Au).

In situ observation using TEM is indispensable to clarify the intermediate process of the above morphological transformation from the multilayer to Au nanoparticles in real space. Upon heating, the layered structure of Au was changing gradually. At 393 K as shown in Figure 3(a), roughness at the boundary between a-Au and a-Si occurred partially on the atomic scale. The modulation of the boundaries in the multilayer implies that Au diffusion might be activated thermally below the MIC temperature (= 450 K). By further heating above, zigzag patterns were observed at 478 K (=T_zig) with losing the flatness and smoothness (Figure 3(b)). We suppose that, above_, much Au diffusion is promoted on the mesoscopic scale. Consequently, the initial layered structure of a-Au might be modified into the zigzag one by mass transfer of Au. At 508 K, the metastable zigzag pattern in the thin film vanished drastically (Figure 3(c)). Simultaneously, some dynamic fluctuations of Au nanoparticles were probed by the in situ TEM observation. The nano-heterogeneity is characterized by its ambiguous boundary and dynamic motion. The nearly static zigzag pattern at 478 K and slow dynamic nano-he- terogeneity at 508 K (second order) are regarded as a precursor phenomenon of the Au nanoparticle formation.

Both the as-deposited Si and Au layers is an amorphous state, which is determined by XRD (Figure 4(a)). In fact, there is no sharp Bragg spot in the 2D X-ray diffraction except for those from the substrate. The weak spots are from the substrate with different wavelengths such as Cu Kβ₁ (λ = 0.1392 nm) and W Lα₁ (λ = 0.1476 nm) due to filament of X-ray generator. Judging from no Debye rings, the as-deposited multilayer in Au_1−xSi_x consists only of a-Au and a-Si. In the sample preparations, Au atoms cannot form crystal in the multilayer during deposition, since t_Au (=0.8 nm) might be too thin to achieve the critical size for Au crystal nucleation. It is confirmed from a previous study using bilayered c-Au/a-Si (t_Au = 30 nm) [6] , that the Au thickness is sufficient for the crystal nucleation of Au.

On the other hand, after cycle 4 (T_max = 1073 K), Au crystallization was distinguished by the Debye rings on

the XRD pattern (Figure 4(b)). From the peak widths, the average size of c-Au nanoparticles was estimated at room temperature and was determined to be 3.0 nm in radius. The ideal Debye rings of the Bragg reflections in Figure 4(b) suggest that the Au nanoparticles have no orientational order in the thin film. Thus, after annealing, the c-Au nanoparticles were observed at random sites by TEM and random crystal orientations by XRD. Therefore, XRD data does not contradict with the TEM observations relating to the morphological transformation from multilayer to Au nanoparticles. To compare the Au crystal with Si one, the observed 2D data were reduced into 1D (Figure 4(c) and Figure 4(d)). Moreover, after cycle 4, metastable Au silicides such as Au₂Si, Au₅Si₂ and Au₇Si [8] -[10] were not seen at room temperature (Figure 4(b)). Metastable silicides were bit seen during in situ XRD observations. By Williamson-Hall method [26] , local root-mean-square strain of c-Au was analyzed using the thermal cycling sample. From the peak widths of Bragg reflections in c-Au (Figure 4(d)), strain of the c-Au nanoparticles is negligibly small.

Figures 5(a)-(d) show the respective integrated intensities of Si 111 Bragg reflection at each thermal cycle. At cycle 1, a-Si crystallized partially and irreversibly at around 450 K (=) (Figure 5(a)). This is regarded as typical MIC, but no Au crystallization was detected. Once the MIC process occurred, the volume of c-Si was almost constant over all thermal cycles (cycles 1 - 4). The irreversible Si crystallization can be interpreted as frozen Si atoms, which are unstably captured in the multilayer, whereas the as-deposited a-Si is partially stabilized by crystallization. After cycle 4, XRD in Figure 4(b) and Figure 4(d) reveal that the Bragg intensities of c-Au are larger than those of c-Si in spite of t_Au < t_Si. For quantitative analysis of each crystal volumes, we estimate the crystal volume ratio by X-ray kinematical theory [27] . Generally, the scattering volume, V, in XRD is proportional to integrated intensities of Bragg reflections. In the case of polycrystal, the relation is given by,

is the structure factor and e^−2M is the Debye-Waller factor as thermal dumping function. LP is the Lorentz-polarization factor. is the multiplicity factor for polycrystalline. Here, we ignore the extinction effect, since multiple scattering is not enhanced in the nanoparticles. Also, X-ray absorption of c-Au is negligible

due to quite small nanoparticles. When the observed integrated intensities of Bragg reflections are normalized using Equation (2), the substantial volumes ratio of V^Si (c-Si) to V^Au (c-Au) is evaluated to be 1.07. If all a-Au crystallize, V^Au of c-Au becomes 16%. Using the crystal volume ratio, the volume of c-Si is calculated to be 17.2% (Table 1). The rest of Si (66.8%) still remains frozen as the amorphous state even after cycle 4. Hence, a-Si might be stable trivially in the thin film. In addition to small amount of c-Si on the MIC process, lower (=450 K) was observed in this study. Previously, on heating (3 K/min), the MIC occurred at around 483 K with decreasing Bragg intensity of c-Au in the thin film (30 nm of c-Au on a 200 nm thick a-Si) [6] . Moreover, large grained c-Si was obtained in the study. Therefore, the partial and irreversible MIC at cycle 1 in this study is entirely different from the conventional MIC process. Here, we emphasize that an initial Au state (a-Au or c-Au) influences extensively on the MIC process in a-Si.

On heating the film at cycle 2 (T_max = 673 K), a-Au started to crystallize at around 480 K (=T_c). Figure 6(a) shows temperature dependence of intensity of Au 111 Bragg reflection. The crystallization temperature of Au coincides with T_zig. In real space, zigzag patterns of Au layers in Figure 3(b) were formed at the same temperature as a-Au started to crystallize. A significant point is that Au nucleation process has some relation with the morphological changes. From the mass transfer of Au, which is explained by shape change from flat stripe to zigzag, Au diffusion might assist the crystal nucleation. The intensity of Au 111 Bragg reflection gradually increased with increasing temperature. The previous study [16] suggests that the morphological transformation from the layered a-Au to the c-Au nanoparticles takes place irreversibly. In contrast to the MIC in a-Si, it is emphasized that the whole volume of a-Au can transform to c-Au at around 560 K. On deposition process of Au, the confinement in the thin Au layer (t_Au = 0.8 nm) below the critical size for crystal nucleation enforces the atomic arrangement to form unstable amorphous, which is far from an equilibrium state. Consequently, all a-Au can transform to crystal, since the driving force from a-Au to c-Au is large enough. Further heating at cycle 2 caused the eutectic Au_1−xSi_x to melt. Above 570 K, the intensity of c-Au decreased and disappeared completely at 600 K (=), although the Bragg intensities of c-Si were almost constant. Thus, it is interpreted that liquid AuSi (ℓ-AuSi) was formed with a mass transfer from a-Si (Table 1). Here, the bulk eutectic point () in Au_1−xSi_x is 636 K. In the same manner with low, the thin film has a tendency of low. On another approach, temperature dependence of full width at half maximum (FWHM) of Au 111 Bragg reflection is calculated by profile fitting using the asymmetric pseudo-Voigt function (Figure 6(b)). Remarkably, FWHM at the early stage of crystallization became large reflecting on small nuclei from a-Au. Accompanying with coarsening up to the c-Au nanoparticles, FWHM of the Bragg reflection became smaller. At the temperature range from 540 K up to 600 K, constant FWHM implies that the nanoparticle size of c-Au was suppressed at 2.6 nm in radius (r^obs), which is determined by the conventional Scherrer equation [28] .

Corresponding to the early stage of Au crystallization and the morphological change including diffusion process,

electrical conductivity (s) was measured separately using other as-deposited sample at the same heating rate (10 K/min) in Figure 6(c). At around T_c of c-Au, gradient of the conductivity as a function of temperature varied a little. Since s is expressed by mobility of species having charge, it is adequate to consider thermally induced local diffusion of Au atoms above T_c (~T_zig). The abrupt increase of s above might be originated from free-Au migration in the liquid state. In order to interpret the irreversible morphological transfer and the slow crystal growth process of Au, we assume the process by introducing the conventional “diffusion-limited model” and “interface-limited model”. Just below Au crystallization temperature, Au diffusion occurs on atomic scale in amorphous multilayer. By the TEM observations, Au diffusion-driven zigzag structure supports the Au mass transfer. At the early stage of crystallization, Au atoms aggregate each other and crystal nanoparticle is formed. At the late stage of crystallization, interface-limited model is effective once size of nanoparticle exceeds the critical size. However, repulsive interaction near the eutectic point suppress the further growth of Au nanoparticle.

We focus the early stage of Au diffusion process. The local diffusion is described by slow dynamics has less geometrical constraint, that is, shape flexibility, compared with the interface-limited process. Then, we introduce the diffusion-assisted nucleation model [29] [30] , which is provided by,

where k_B is the Boltzmann constant and DG is the free energy of an n-monomer embryo. P is the size distribution function, which is represented by n-monomer embryo. c is the monomer concentration. In the Smoluchowski expression, K_n is approximately obtained by,

Here, D is diffusion coefficient for monomers and a is radius of the monomer. In the case of a-Au/a-Si system above T_c, D in Equation (4) corresponds to Au atomic diffusion into a-Si. Experimentally, the activation energy of Au atomic diffusion inside a-Si was obtained in the previous study [31] . The effective radius of the embryo, r, is expressed as an^1/3 considering spherical aggregation of n-monomer [29] . In the XRD experiments, we obtained temperature dependence of V^Au(T) under the constant heating rate, DT/Dt (= 10 K/min). If we assume that

the growth curve, V^Au(T), has a similarity with time dependence of the volume, V^Au(t), is

approximately obtained. With satisfying the diffusion-assisted nucleation, the Equation (3) is modified by,

As mentioned earlier, Bragg intensity from c-Au (Figure 6(a)) is proportional to V^Au(T). The first derivative

of the observed V^Au(T), , is easily calculated. Also, r^obs(T) is estimated by FWHM of Bragg reflec-

tion (Figure 6(b)). We used D(T) in the previous study [31] . Unknown parameters are DG and c(T) in Equation (5). From 480 to 600 K, we assume that DG in Equation (5) is approximately expressed by a weak linear relation with temperature (). Also, it is assumed that c(T) is described by proper function such as tanh

. By referring, the self-consistent calculation can determine the DG and c(T) simul-

taneously. As a result, the calculated growth curve including diffusion process is exhibited by the solid curve in Figure 6(a). Above 560 K, actual intensity of Au 111 Bragg reflection decreased due to melting. Finally, both experimental results and model calculation supports the idea based on the diffusion-assisted nucleation process of Au. We notice that the initial state of the a-Au/a-Si thin film has major influence on both the irreversible MIC in Si and the Au nano-crystallization. In fact, the relatively slow diffusion of Au was enhanced in a-Si compared with Au diffusion in c-Si [31] .

If irreversible morphological transformation from the multilayer to Au nanoparticle occurs, the reversible liquid-solid phase transition was observed repeatedly. Even above of c-Au, the whole thin film does not melt, since c-Si and a-Si existed at T_max and the flat surface in the thin film was maintained at room temperature after thermal cycling. Heterogeneous ℓ-AuSi in the thin film above is the key to understand the reversible phase transition between the ℓ-AuSi and the c-Au nanoparticles. On cooling at cycle 2, T_c was determined to be 556 K by temperature dependences of intensity and FWHM of Au 111 Bragg reflection (Figure 7(a) and Figure 7(b)). Since small nuclei were not observable in small scattering volume of the thin film, Bragg intensities were derived not from small nuclei, but from the c-Au nanoparticles. A more significant point is that FWHM of the Bragg reflection is invariant on cooling process in Figure 7(b). It is well known that the constant FWHM of Bragg reflection means non-coarsening of a crystal domain [32] . Hence, the population of c-Au nanoparticles is increasing on cooling without crystal domain growth. The behavior is analogous with reversible spin ordering in a ferromagnet. Therefore, we apply the Ising model [33] into the crystallization from the ℓ-AuSi to the c-Au nanoparticles. The feature of the Ising model is that the Hamiltonian is described only by the first nearest neigh- bor interaction. Based on the Ising model, crystallization process is provided by,

where β is the critical exponent. By the least square fitting, the experimentally obtained β^obs is evaluated to be 0.17. The fitted growth curve is represented by the solid line in Figure 6(a). The exact solution of the 2D Ising model showed the β = 1/8 (=0.125) [33] . In order to investigate the crossover from 2D to 3D property in the Is- ing model, systematic simulations in the thin film [34] were demonstrated, where periodic boundary condition is employed on the x-y planes. Here, the x-y plane is parallel to the substrate and the z direction is perpendicular to

it. In the simulation box denoted by L × L × N_z, the critical exponent β^cal becomes 0.17, when L is 160 and N_z is 6. The accordance between β^obs and β^cal indicates that the nanoparticles without their aggregations are consistent with the ideal spin system with finite thickness along the stacking sequence direction. Also, it is supposed that the actual appearance of the Au nanoparticles obeys short-range order (SRO) interaction. The analogy provides a significant insight into the reversible ℓ-AuSi-c-Au nanoparticle phase transition. Figure 7(c) reveals electrical conductivity on cooling. Vanishing of ℓ-AuSi causes the decrease of the conductivity at around T_c. Below T_c, the conductivity value became smaller than that on heating as shown in Figure 6(c), which is represented by the irreversible transformation from the multilayer to nano particle. At room temperature, the difference between them is originated from the layered a-Au land the isolated Au nanoparticles.

As mentioned earlier, the reversible liquid-solid phase transition in the thin film occurs between heterogeneous ℓ-AuSi and c-Au nanoparticles. When we consider the surface and bulk phases at the equilibrium state, the surface tension in binary liquid alloy should be treated thermodynamically. Recently, it is predicted from a thermodynamical perspective, that spherical particle size [19] [35] is obtained by introducing the surface tension [36] . The relationships between the observed and radius of the c-Au nanoparticles (r^obs) were calculated using the basic theory in nano-sized binary alloy system [19] [35] . In in situ XRD, r^obs in cycle 2 - 4 is obtained by the Scherrer equation as listed in Table 2 [28] . While, r^obs is also determined by direct TEM observations. In fact, the particle size of c-Au has a relation with as shown in Figure 8.

In the theory [19] [35] , the spherical particle having radius, r, is connected with in binary alloy system. Gibbs energy of bulk () and surface () in the binary system is given by,

Here, is Gibbs energy difference between liquid and solid in pure system (i-component: i = 1 or 2), while is the excess Gibbs energy of liquid phase in binary system. Surface tension of pure solid is expressed by. s^L reveals the surface tension of binary liquid combined with molar volume of the binary liquid, V^L. s^L is numerically calculated using and. V^L is defined by. corresponds to the molar volume for the pure liquid component. In contrast, and are surface tension and molar volume of pure solid, respectively. It should be noticed that, , and contain the temperature factor of (-T_m,i). T_m,i is the melting point in pure system i. If the condition of is satisfied, Equation (8) is modified as,

Using the temperature factor and Equation (9), the at fixing concentration x₁ (=1 − x₂) is provided depending on the Au nanoparticle radius, r. At x_E = 0.186 in AuSi system, the calculated radius of the Au nanoparticle, r^cal, is numerically obtained using each (Table 2). Here, in the calculations, thermodynamic data and parameters in Au-Si system as previously reported paper [35] are used. Experimentally, ℓ-AuSi, compositional fluctuations occur naturally. If we permit the possible compositional fluctuations, the r^obs obtained both from XRD and TEM is in good agreement with the r^cal within experimental errors (Figure 8). Consequently, in the thin film, the lowering eutectic point of the Au nanoparticle is well explained theoretically.

The scenario of two different irreversible/reversible transformations is illustrated in Figure 9. The irreversible transformation from the layered a-Au to c-Au nanoparticles is distinguished from the reversible liquid-nano- crystal transformation. MIC process of a-Si partially occurred from the as-deposited sample (). The partial crystallization of Si and the constant volume of Si over all thermal cycling reflect on little free energy difference between a-Si and c-Si in the specific thin film. The nucleation sites might be distorted positions in the vicinity of the Al₂O₃ substrate or the boundaries between a-Au and a-Si. The layered a-Au could be modified on

atomic scale by crystal nucleation of Si, however, crystal nucleation of Au is not induced at this stage.

Next step on heating is that zigzag structure and Au crystallization occurs simultaneously as a precursor phenomenon of Au nanoparticle formation (T > T_c - T_zig). As a result of aggregation of diffusing Au atoms, Au nucleation takes place on the off-sites from the original a-Au layers. Remarkably slow temperature evolution of V^Au(T) has no discrepancy with time consuming diffusion-driven nucleation. At the same time, shape transfer based on the diffusion-limited model happens in low energy barrier. When the interface-limited process occurs in the morphological transformation, large strain and boundary energy should be considered additionally. In spite of the diffusion-limited motion, all of a-Au is crystallized due to large driving force between amorphous and crystal.

Focusing on the reversible transformation, heterogeneous distribution of a-Si (V_Si—66.8%) at T_max prevents to melt the whole film (T > T_m). Partially melted ℓ-AuSi region distributes independently in the center of c-Au before melting. On cooling, crystal nuclei of Au appear inside the isolated regions. With excluding Si atoms, the nuclei develop to the c-Au nanoparticles once those exceed over the critical size. Inside the ℓ-AuSi, the nanoparticle size of c-Au is governed by the surface tension. The size of c-Au is optimized by a competition between bulk and surface free energy. The space restriction of the confined ℓ-AuSi avoids aggregation between the c-Au nanoparticles. Then, the same reversible phase transition between the ℓ-AuSi and the c-Au nanoparticles occurs repeatedly. Also, less degrees of freedom in the crystallization mechanism leads to the ideal Ising model described only by simple SRO correlation. The idealized spin system is applied into the spontaneous nucleation and size-limited growth in the thin film.